Bayesian $L_{\frac{1}{2}}$ regression

It is well known that bridge regression enjoys superior theoretical properties than traditional LASSO. However, the current latent variable representation of its Bayesian counterpart, based on the exponential power prior, is computationally expensive in higher dimensions. In this paper, we show that the exponential power prior has a closed-form scale mixture of normal decomposition for $\alpha=(\frac{1}{2})^\gamma, \gamma \in \mathbb{N}^+$. We develop a partially collapsed Gibbs sampling scheme, which outperforms existing Markov chain Monte Carlo strategies, we also study theoretical properties under this prior when $p>n$. In addition, we introduce a non-separable bridge penalty function inspired by the fully Bayesian formulation and a novel, efficient, coordinate-descent algorithm. We prove the algorithm's convergence and show that the local minimizer from our optimization algorithm has an oracle property. Finally, simulation studies were carried out to illustrate the performance of the new algorithms.